Small Noise Asymptotics for a Stochastic Growth Model

We develop analytic asymptotic methods to characterize time series properties of nonlinear dynamic stochastic models. We focus on a stochastic growth model which is representative of the models underlying much of modern macroeconomics. Taking limits as the stochastic shocks become small, we derive a functional central limit theorem, a large deviation principle, and a moderate deviation principle. These allow us to calculate analytically the asymptotic distribution of the capital stock, and to obtain bounds on the probability that the log of the capital stock will differ from its deterministic steady state level by a given amount. This latter result can be applied to characterize the probability and frequency of large business cycles. We then illustrate our theoretical results through some simulations. We find that our results do a good job of characterizing the model economy, both in terms of its average behavior and its occasional large cyclical fluctuations.

Bayesian and adaptive optimal policy under model uncertainty

We study the problem of a policymaker who seeks to set policy optimally in an economy where the true economic structure is unobserved, and policymakers optimally learn from their observations of the economy. This is a classic problem of learning and control, variants of which have been studied in the past, but little with forward-looking variables which are a key component of modern policy-relevant models. As in most Bayesian learning problems, the optimal policy typically includes an experimentation component reflecting the endogeneity of information. We develop algorithms to solve numerically for the Bayesian optimal policy (BOP). However the BOP is only feasible in relatively small models, and thus we also consider a simpler specification we term adaptive optimal policy (AOP) which allows policymakers to update their beliefs but shortcuts the experimentation motive. In our setting, the AOP is significantly easier to compute, and in many cases provides a good approximation to the BOP. We provide a simple example to illustrate the role of learning and experimentation in an MJLQ framework. JEL Classification: E42, E52, E5

Modeling Model Uncertainty

Recently there has been a great deal of interest in studying monetary policy under model uncertainty. We point out that different assumptions about the uncertainty may result in drastically different robust' policy recommendations. Therefore, we develop new methods to analyze uncertainty about the parameters of a model, the lag specification, the serial correlation of shocks, and the effects of real time data in one coherent structure. We consider both parametric and nonparametric specifications of this structure and use them to estimate the uncertainty in a small model of the US economy. We then use our estimates to compute robust Bayesian and minimax monetary policy rules, which are designed to perform well in the face of uncertainty. Our results suggest that the aggressiveness recently found in robust policy rules is likely to be caused by overemphasizing uncertainty about economic dynamics at low frequencies.

Modeling model uncertainty

Recently there has been much interest in studying monetary policy under model uncertainty. We develop methods to analyze different sources of uncertainty in one coherent structure useful for policy decisions. We show how to estimate the size of the uncertainty based on time series data, and incorporate this uncertainty in policy optimization. We propose two different approaches to modeling model uncertainty. The first is model error modeling, which imposes additional structure on the errors of an estimated model, and builds a statistical description of the uncertainty around a model. The second is set membership identification, which uses a deterministic approach to find a set of models consistent with data and prior assumptions. The center of this set becomes a benchmark model, and the radius measures model uncertainty. Using both approaches, we compute the robust monetary policy under different model uncertainty specifications in a small model of the US economy. JEL Classification: E52, C32, D81estimation, Model uncertainty, monetary policy

Optimal monetary policy under uncertainty: a Markov jump-linear-quadratic approach

This paper studies the design of optimal monetary policy under uncertainty using a Markov jump-linear-quadratic (MJLQ) approach. To approximate the uncertainty that policymakers face, the authors use different discrete modes in a Markov chain and take mode-dependent linear-quadratic approximations of the underlying model. This allows the authors to apply a powerful methodology with convenient solution algorithms that they have developed. They apply their methods to analyze the effects of uncertainty and potential gains from experimentation for two sources of uncertainty in the New Keynesian Phillips curve. The examples highlight that learning may have sizable effects on losses and, although it is generally beneficial, it need not always be so. The experimentation component typically has little effect and in some cases it can lead to attenuation of policy.Monetary policy ; Econometric models

Optimal Monetary Policy Under Uncertainty in DSGE Models: A Markov Jump-Linear-Quadratic Approach

We study the design of optimal monetary policy under uncertainty in a dynamic stochastic general equilibrium model. We use a Markov jump-linear-quadratic (MJLQ) approach to study policy design, proxying the uncertainty by different discrete modes in a Markov chain, and by taking mode-dependent linear-quadratic approximations of the underlying model. This allows us to apply a powerful methodology with convenient solution algorithms that we have developed. We apply our methods to a benchmark new-Keynesian model, analyzing how policy is affected by uncertainty, and how learning and active testing affect policy and losses.

Bayesian and Adaptive Optimal Policy under Model Uncertainty

We study the problem of a policymaker who seeks to set policy optimally in an economy where the true economic structure is unobserved, and policymakers optimally learn from their observations of the economy. This is a classic problem of learning and control, variants of which have been studied in the past, but little with forward-looking variables which are a key component of modern policy-relevant models. As in most Bayesian learning problems, the optimal policy typically includes an experimentation component reflecting the endogeneity of information. We develop algorithms to solve numerically for the Bayesian optimal policy (BOP). However the BOP is only feasible in relatively small models, and thus we also consider a simpler specification we term adaptive optimal policy (AOP) which allows policymakers to update their beliefs but shortcuts the experimentation motive. In our setting, the AOP is significantly easier to compute, and in many cases provides a good approximation to the BOP. We provide a simple example to illustrate the role of learning and experimentation in an MJLQ framework.Optimal Monetary Policy, Learning, Recursive Saddlepoint Method

Generalized Stochastic Gradient Learning

We study the properties of generalized stochastic gradient (GSG) learning in forwardlooking models. We examine how the conditions for stability of standard stochastic gradient (SG) learning both di1er from and are related to E-stability, which governs stability under least squares learning. SG algorithms are sensitive to units of measurement and we show that there is a transformation of variables for which E-stability governs SG stability. GSG algorithms with constant gain have a deeper justification in terms of parameter drift, robustness and risk sensitivity

Shocks and Government Beliefs: The Rise and Fall of American Inflation

We use a Bayesian Markov Chain Monte Carlo algorithm to estimate a model that allows temporary gaps between a true expectational Phillips curve and the monetary authority's approximating non-expectational Phillips curve. A dynamic programming problem implies that the monetary authority's inflation target evolves as its estimated Phillips curve moves. Our estimates attribute the rise and fall of post WWII inflation in the US to an intricate interaction between the monetary authority's beliefs and economic shocks. Shocks in the 1970s altered the monetary authority's estimates and made it misperceive the tradeoff between inflation and unemployment. That caused a sharp rise in inflation in the 1970s. Our estimates say that policymakers updated their beliefs continuously. By the 1980s, their beliefs about the Phillips curve had changed enough to account for Volcker's conquest of US inflation in the early 1980s.